Genet. Sel. Evol. 34 (2002) 171–192
© INRA, EDP Sciences, 2002
DOI: 10.1051/gse:2002002
171
Original article
Optimal selection on two quantitative trait
loci with linkage
Jack C.M. DEKKERSa∗ , Reena CHAKRABORTYa ,
Laurence MOREAUb
a
Department of Animal Science, 225 Kildee Hall,
Iowa State University Ames, IA, 50011, USA
b
INRA-UPS-INA PG, Station de génétique végétale,
Ferme du Moulon, 91190 Gif-sur-Yvette, France
(Received 5 February 2001; accepted 15 October 2001)
Abstract – A mathematical approach to optimize selection on multiple quantitative trait loci
(QTL) and an estimate of residual polygenic effects was applied to selection on two linked or
unlinked additive QTL. Strategies to maximize total or cumulative discounted response over ten
generations were compared to standard QTL selection on the sum of breeding values for the QTL
and an estimated breeding value for polygenes, and to phenotypic selection. Optimal selection
resulted in greater response to selection than standard QTL or phenotypic selection. Tight linkage
between the QTL (recombination rate 0.05) resulted in a slightly lower response for standard
QTL and phenotypic selection but in a greater response for optimal selection. Optimal selection
capitalized on linkage by emphasizing selection on favorable haplotypes. When the objective
was to maximize total response after ten generations and QTL were unlinked, optimal selection
increased QTL frequencies to fixation in a near linear manner. When starting frequencies were
equal for the two QTL, equal emphasis was given to each QTL, regardless of the difference
in effects of the QTL and regardless of the linkage, but the emphasis given to each of the two
QTL was not additive. These results demonstrate the ability of optimal selection to capitalize
on information on the complex genetic basis of quantitative traits that is forthcoming.
selection / marker-assisted selection / quantitative trait loci / optimization
1. INTRODUCTION
The advent of molecular genetics has opened opportunities to enhance
selection for quantitative traits by incorporating information on the identified
quantitative trait loci (QTL) or on the genetic markers that are linked to QTL
in genetic improvement programs. In what follows, we will refer to identified
QTL as QTL that are in complete population gametic phase disequilibrium
∗
Correspondence and reprints
E-mail: [email protected]
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with the observed genetic polymorphism (i.e. the QTL itself or a tightly linked
marker), in contrast to marked QTL, for which the observable polymorphism
is in linkage equilibrium with the QTL.
The generally accepted strategy for using QTL information in selection,
which will be referred to as standard QTL selection, is to select on a simple
index of an estimate of the breeding value for the identified or marked QTL
(α) and an estimated breeding value (EBV) of the residual polygenic effects
of the individual: I = α + EBV [7,15]. For additive QTL and when the
components of the index are estimated using best linear unbiased prediction
(BLUP) that includes the QTL as a fixed or random effect [8], standard QTL
selection maximizes the response from the current to the next generation.
Standard QTL selection, however, does not maximize the single generation
response for QTL with non-additive effects [4], nor does it maximize the
response over multiple generations, even for additive QTL [9,11,14]. Thus,
the selection emphasis placed on the QTL in standard QTL selection, which
is determined by the QTL breeding value, is not optimal for many situations.
The methods to optimize the use of a single identified QTL in selection
were developed by Dekkers and van Arendonk [5] and Manfredi et al. [13].
Both methods focused on selection on a single identified QTL. Dekkers and
Chakraborty [6] evaluated the benefit of optimal selection on a single identified
QTL for a range of parameters and found substantial differences between
optimal and standard QTL selection for QTL with overdominance.
With the advances in molecular genetics, multiple QTL are being identified
for many livestock species [1] and, therefore, the information from more than
one QTL must be incorporated in selection procedures. Standard QTL selection
can accommodate multiple QTL by including the sum of breeding values of the
individual QTL. Selection emphasis on an individual QTL is then determined by
the relative magnitude of its breeding value. Hospital et al. [10] investigated
QTL selection strategies in which equal emphasis was given to each QTL
regardless of their effect, with the aim to fix all QTL as rapidly as possible.
They, however, pointed out that it is unclear whether this strategy indeed
minimizes the time to fixation and that optimal strategies for selection on
multiple QTL are lacking.
In a companion paper, Chakraborty et al. [3] extended the method of Dekkers
and van Arendonk [5] to allow optimization of selection on multiple QTL. The
objectives of this paper were to implement this methodology for selection on
two QTL and to evaluate the effect of linkage between QTL on characteristics
of response to optimal QTL selection, in comparison to standard QTL and
phenotypic selection.
Optimal selection on two linked QTL
173
2. METHODS
2.1. Population structure and genetic model
The deterministic model described by Chakraborty et al. [3] was used to
model selection on a trait affected by two identified QTL and residual polygenic
effects in a population with discrete generations. Parameters for the population
and genetic model were chosen to enable investigation and illustration of the
properties of the optimal selection strategy and of the impact of linkage between
QTL, rather than to be representative of practical breeding programs.
Selected fractions were 20% for both males and females. All candidates
were genotyped for the identified QTL prior to selection and the effect and
parental origin of each QTL allele was assumed known. Polygenic effects
were modeled following the infinitesimal genetic model [2] but with constant
variance. Both QTL were additive and had two alleles (A1 , A2 and B1 , B2 ).
Bilocus genotypes for the QTL are coded as, e.g., A1 A2 B1 B2 , where the first
allele for each locus designates the paternal allele. Starting frequencies of the
favorable alleles (A1 and B1 ) were 0.1 for both QTL and the starting population
was in gametic phase equilibrium.
Allele substitution effects were equal to 1 and 1.5 in units of standard
deviations (σ) of polygenic EBV, which is the unit of interest when comparing
alternative QTL selection strategies [5]. For a trait with heritability equal
to 0.25, these convert to allele substitution effects of 0.5 and 0.75 in units of
standard deviations of true polygenic effects and to 0.25 and 0.375 in phenotypic
standard deviation units if polygenic EBV are based on own phenotype only [6].
To evaluate the impact of linkage, recombination rates (r) of 0.50 (unlinked),
0.20, and 0.05 between the QTL were considered. A situation of a single QTL
with an additive effect equal to the sum of the effects of the two QTL, i.e. 2.5σ,
was also considered. Note that this is identical to a zero recombination rate
between the QTL but with complete gametic phase disequilibrium in the base
population, such that positive alleles are always in a coupling phase.
2.2. Selection methods and objectives
Optimal QTL selection was compared to standard QTL selection and to
phenotypic selection. Selection was on the following general index [5]:
Iijmt = θmt + (EBV ijmt − BV mt )
(1)
where θmt is the value assigned to individuals of the QTL genotype m in
generation t, which differed by selection strategy, and EBV ijmt is the polygenic
EBV of animal i, which is deviated from the mean polygenic BV of individuals
of genotype m in generation t (BV mt ). Selection on index (1) was modeled
by truncation selection across distributions of EBV by genotype, with means
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equal to θmt and standard deviation equal to σ. Given θmt and σ, proportions
selected from each distribution were derived by bisection and were used to
model changes in QTL frequencies and polygenic means, as described by
Chakrabory et al. [3].
For phenotypic selection, means θmt were derived by regressing the mean
phenotypic value of each genotype (= qm + BV mt where qm is the sum of
genotypic values for the QTL) towards an overall mean based on heritability [5,
6], which was set equal to 0.25. For standard QTL selection, means θmt were
set equal to αm + BV mt , where αm is the sum of breeding values for the two
QTL based on allele substitution effects [5,6].
The methodology developed by Chakraborty et al. [3] was used to derive
optimal QTL selection strategies. Standard QTL selection provided the starting
point for the iterative solution process. Optimal QTL selection strategies were
obtained for two alternative objectives: (1) maximizing cumulative response
after ten generations and (2) maximizing cumulative discounted response over
ten generations. These will be referred to as optimal terminal and optimal
discounted QTL selection,
respectively. Cumulative discounted response was
computed as R = Tt=0 wt Gt where Gt is the mean total genotypic value in
generation t, including both the QTL and polygenes, and wt is a discount factor,
which was computed based on an interest rate of ρ = 0.1 per generation as:
wt = 1/(1 + ρ)t . Note that the terminal response is given by G10 .
The optimization program provides optimal fractions to select from each
genotype in each generation. Based on the standard normal distribution theory,
these fractions can be used to derive the standardized truncation point that
is associated with genotype m in generation t, xmt . Following Dekkers and
van Arendonk [5], differences in truncation points between genotypes were
translated to the differences between means θmt that are assigned to each
genotype in index (1) as:
θmt − θref,t = (xref,t − xmt )σ
(2)
where ref refers to an arbitrary reference genotype. Here, genotype A1 A2 B1 B2
was used as the reference genotype. The means derived by (2) quantify the
emphasis that is put on each QTL genotype in index (1) under optimal selection.
These will be referred to as optimal genotype values, in contrast to standard
genotypic values (qm), which refer to the impact of QTL genotype on phenotype.
Optimal genotype values computed by equation (2) apply to the bilocus QTL
genotype. To quantify the emphasis that is placed on individual QTL, single
locus genotype values were computed by averaging optimal bilocus genotype
values over genotypes for the other locus. For example, the mean optimal
genotype value for A1 A1 was obtained as the average of the optimal bilocus
genotype values for genotypes that included A1 A1 , weighted by the frequency
of each genotype.
Optimal selection on two linked QTL
175
2.3. Gametic phase disequilibrium
Selection generates gametic phase disequilibrium between QTL and between
the QTL and polygenes [2] (gametic phase disequilibrium among polygenes
was ignored by assuming constant polygenic variance). Denoting Pij as the
frequency of QTL haplotype ij, and Pi the frequency of allele i, gametic phase
disequilibrium between the two QTL was computed following Lewontin [12]
as:
D = (PA1 B1 PA2 B2 − PA1 B2 PA2 B1 )/Dmax
(3)
where
Dmax = min(PA1 PB1 , PA2 PB2 ) when D < 0
and
Dmax = min(PA1 PB2 , PA2 PB1 ) when D > 0.
Gametic phase disequilibrium between the QTL and polygenes was quantified
by the correlation of standard genotypic values for the QTL (qm ) with polygenic
breeding values as:
cov(qm , BV mt )
rt = √
2
var(qm ) var(BV mt ) + σpol
(4)
2
where σpol
is the polygenic variance within QTL genotype. Variances and
covariances of qm and BVmt were computed based on haplotype means weighted
by haplotype frequencies.
3. RESULTS
3.1. Response to selection
3.1.1. Maximizing terminal response
Cumulative responses over ten generations in population average QTL,
polygenic, and total genotypic values are listed in Table I for the alternative
selection strategies and genetic models. Polygenic responses were based on
mean polygenic breeding values and QTL responses on mean genotypic values
for the two QTL, both deviated from the genetic level in the starting population.
As expected, optimal terminal QTL selection resulted in the greatest mean
total genotypic value in the final generation (Tab. I), followed by phenotypic
selection and then standard QTL selection. The greater terminal response for
phenotypic than standard QTL selection agrees with previous studies, which
investigated selection on a single identified QTL [9,11,14]. The difference
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Table I. Terminal responses over ten generations (in phenotypic standard deviations)
to phenotypic, standard QTL, and optimal QTL selection with one or two QTL with
a varying recombination rate (r). QTL substitution effects are 1.0 and 1.5 standard
deviations of polygenic EBV for two QTL and 2.5 for one QTL. Polygenic heritability
is 0.25 and starting frequencies of favorable alleles are equal to 0.1 for all QTL.
In brackets, total responses are also expressed relative to response for phenotypic
selection.
Selection
strategy
Response
component
Optimal
Total
Polygenic
QTL
Total
Polygenic
QTL
Total
Polygenic
QTL
Standard
Phenotypic
Two QTL
r = 0.05
4.31 (103.8)
3.20 (97.3)
1.11 (128.7)
3.97 (95.5)
2.84 (86.4)
1.12 (130.4)
4.15
3.29
0.86
r = 0.20
4.27 (102.6)
3.18 (96.6)
1.09 (125.4)
3.97 (95.5)
2.85 (86.6)
1.12 (128.9)
4.16
3.29
0.87
One QTL
r = 0.50
4.27 (102.4)
3.18 (96.6)
1.09 (123.7)
3.98 (95.5)
2.86 (86.9)
1.12 (127.6)
4.17
3.29
0.88
4.44 (101.7)
3.32 (102.1)
1.12 (100.3)
4.10 (93.9)
2.98 (91.7)
1.13 (100.4)
4.37
3.25
1.12
between phenotypic and standard QTL selection was, however, slightly less
for two QTL than when their effects were concentrated into one QTL (−4.5%
for two QTL compared to −6.1% for one QTL). Phenotypic selection resulted
in a similar QTL response than standard QTL selection with a single QTL but
in substantially less QTL response with two QTL. Lost polygenic response for
standard QTL selection relative to phenotypic selection was, however, smaller
for one QTL than for two QTL.
Extra terminal response from optimal over phenotypic selection was greatest
for two tightly linked QTL (r = 0.05) (+3.8%) and smallest for the one QTL
case (+1.7%) (Tab. I). With two QTL, the greater QTL response was the main
reason for the extra response from optimal over phenotypic selection; polygenic
response was slightly lower for optimal than for phenotypic selection. This is in
contrast to the one QTL case, where greater polygenic response was responsible
for the greater total response for optimal over phenotypic selection and QTL
responses were similar between the two strategies. Note that the one QTL
cases is equivalent to the two QTL case with complete linkage and complete
disequilibrium. Thus, the one QTL case is not an extension to complete linkage
of the two QTL cases examined here, for which the QTL were in equilibrium
in the base population.
Extra response from optimal over standard QTL selection increased with
linkage from +6.9% for unlinked loci to +8.3% for tightly linked QTL (Tab. I).
Optimal selection on two linked QTL
177
With one QTL, extra response from optimal selection was intermediate to
these values at +7.8%. Extra polygenic response was the main reason for
the difference in total response between optimal and standard QTL selection.
This confirms the previous finding for the one QTL case by Dekkers and
van Arendonk [5] that optimal terminal QTL selection maximizes polygenic
response over generations while achieving near fixation of the QTL at the end
of the planning horizon.
Tight linkage between the QTL resulted in slightly greater total response
for optimal selection compared to moderate or no linkage (4.31 vs. 4.27,
Tab. I). In contrast, linkage slightly reduced total response for standard QTL
and phenotypic selection. Although differences were small, this illustrates the
ability of the optimal strategy to capitalize on linkage. This will be further
addressed later in the results section.
Trends in cumulative responses to selection are illustrated in Figure 1 for
standard and optimal QTL selection for two unlinked (Fig. 1a) and tightly linked
(Fig. 1b) QTL. Standard QTL selection reached the maximum QTL genotypic
value after seven generations. Increases in QTL values were more gradual
for optimal terminal QTL selection but also nearly reached their maximal
value by generation ten. Linkage had limited effects on QTL responses.
Because of greater emphasis on the QTL, standard QTL selection resulted in a
lower polygenic response in the initial generations and in a lower cumulative
polygenic response over the planning horizon. This is similar to the results
described previously for a single QTL [5].
3.1.2. Maximizing cumulative discounted response
Comparisons between the three selection strategies when the objective was
to maximize cumulative discounted response over ten generations are shown
in Table II. As expected, optimal QTL selection resulted in the greatest
response for all cases. However, in contrast to the terminal response, cumulative discounted responses were greater for standard QTL than for phenotypic
selection when two QTL were present, because of greater QTL responses in
early generations (Fig. 1). Standard QTL and phenotypic selection had similar
cumulative discounted responses for the one QTL case, with lower responses
for standard QTL selection in later generations offsetting early gains, and losses
in discounted polygenic response offsetting gains in QTL discounted response.
The benefit of optimal over phenotypic selection was greater for cumulative
discounted response (Tab. II) than for terminal response (Tab. I) and greater
for two than for one QTL (Tab. II). The discounted polygenic response was
lower for optimal QTL than for phenotypic selection but this was more than
offset by the extra discounted QTL response. The extra cumulative discounted
response from optimal over phenotypic selection was the greatest for tightly
linked loci but did not increase consistently with degree of linkage. The impact
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18
(a) Unlinked QTL (r = 0.5)
Total
16
14
Polygenes
Genetic Gain
12
10
8
6
4
QTL
2
0
0
1
2
3
4
5
6
7
8
9
10
Generation
18
(b) Linked QTL (r = 0.05)
16
Total
14
Genetic Gain
12
Polygenes
10
8
6
4
QTL
2
0
0
1
2
3
4
5
6
7
8
9
10
Generation
Figure 1. Cumulative gains in polygenic, QTL, and total genotypic means for standard
(- - ◦ - -), optimal terminal ( • ), and optimal cumulative discounted ( ❇ )
selection for (a) two unlinked or (b) two tightly linked QTL (r = 0.05). Starting
frequencies are 0.1 for both QTL.
of linkage on relative and absolute discounted total response was, however,
small, with greater discounted polygenic response with tight linkage partially
offset by lower discounted QTL response (Tab. II).
Extra response from optimal over standard QTL selection was smaller for
cumulative discounted (Tab. II) than for terminal response (Tab. I) because
of the emphasis on early gains in the former objective. Extra cumulative
discounted response from optimal discounted over standard QTL selection was
Optimal selection on two linked QTL
179
Table II. Cumulative discounted responses over ten generations (in phenotypic standard deviations) to selection with one or two QTL, with varying recombination rate (r),
for phenotypic, standard QTL, and optimal QTL selection, when the objective is to
maximize cumulative discounted response. QTL substitution effects are 1.0 and 1.5
standard deviations of polygenic EBV for two QTL and 2.5 for one QTL. Polygenic
heritability is 0.25 and starting frequencies of favorable alleles are equal to 0.1 for
all QTL. Interest rate is 10% per generation. In brackets, total responses are also
expressed relative to response for phenotypic selection.
Selection
strategy
Response
component
Optimal
Total
Polygenic
QTL
Total
Polygenic
QTL
Total
Polygenic
QTL
Standard
Phenotypic
Two QTL
r = 0.05
12.7 (106.0)
8.4 (87.2)
4.3 (182.2)
12.3 (102.5)
7.4 (76.6)
4.9 (207.9)
12.0
9.6
2.4
r = 0.20
12.6 (105.3)
7.9 (82.5)
4.7 (197.3)
12.3 (102.7)
7.4 (76.6)
5.0 (207.8)
12.0
9.6
2.4
One QTL
r = 0.50
12.7 (105.6)
7.9 (82.3)
4.8 (198.8)
12.4 (103.0)
7.4 (76.7)
5.0 (208.3)
12.0
9.6
2.4
14.0 (103.3)
8.5 (92.3)
5.5 (127.0)
13.4 (99.3)
7.3 (79.5)
6.1 (142.2)
13.5
9.2
4.3
slightly greater for the one QTL case (+4.0%) than for two tightly linked QTL
(+3.5), and the smallest (+2.6%) for two QTL with moderate or no linkage.
Optimal discounted QTL selection resulted in substantially greater discounted
polygenic response than standard QTL selection, in particular for one QTL and
for two tightly linked QTL, but this was partially offset by a lower discounted
QTL response.
Figure 1 shows trends in response for optimal discounted QTL selection.
Polygenic and QTL responses for selection that maximized cumulative discounted response were intermediate to those for standard QTL selection and
to selection that optimized terminal responses, but tended to be closer to
trends for the former. For both linked and unlinked QTL, optimal discounted
QTL selection had similar total response as standard QTL selection in early
generations but greater response in later generations. This was achieved by
sacrificing some QTL response in early generations, which allowed an increase
in polygenic response. As a result, optimal selection was able to maintain total
response in early generations, while additional QTL response was capitalized
on in later generations because the QTL were not yet fixed. The amount of
QTL response sacrificed by optimal selection in early generations in favor
of polygenic response was greater for tightly linked QTL than for unlinked
QTL.
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1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0
1
2
3
4
5
6
7
8
9
10
Generation
Figure 2. QTL frequencies for two unlinked QTL with effects equal to 1.0 (◦) and
1.5 (•) standard deviations of polygenic EBV for standard (
), optimal cumulative
discounted (
) and optimal terminal (
for 0.1 starting frequencies and - - - for starting frequencies of 0.3 and 0.1).
3.2. QTL allele frequencies
Figure 2 shows trends in frequencies for two unlinked QTL for standard and
optimal QTL selection. Starting frequencies of the favorable QTL alleles were
0.1. For optimal terminal QTL selection, an initial frequency of 0.3 for the
smaller QTL (A) was also considered.
As expected, standard QTL selection resulted in rapid fixation of the favorable alleles, in particular for the QTL with larger effect. Similar trends, but
with a lower rate of fixation, were observed for optimal QTL selection when
the objective was to maximize cumulative discounted response.
For optimal QTL selection, trends in allele frequencies were nearly linear
when the objective was to maximize terminal response, regardless of starting
frequencies (Fig. 2). When starting frequencies were equal, trends were also
similar for the two QTL, despite a difference in their genotypic effects. When
starting frequencies were different, emphasis on each QTL was such that the
frequencies increased at a nearly constant rate to reach near fixation at the end
of the planning horizon. In contrast, with standard and optimal discounted
QTL selection, rates of fixation were much greater for the QTL with the larger
effect.
Figure 3 illustrates the effect of linkage on trends in QTL frequencies. Only
results for tight linkage (r = 0.05) are shown; those for moderate linkage
(r = 0.2) were intermediate.
181
Optimal selection on two linked QTL
1.0
0.9
0.8
Standard
QTL Frequency
0.7
0.6
0.5
Optimal
0.4
0.3
0.2
0.1
0.0
0
1
2
3
4
5
6
7
8
9
10
Generation
Figure 3. QTL frequencies for two unlinked (- - - -) or tightly linked (r = 0.05)
) QTL with effects equal to 1.0 (◦) and 1.5 (•) standard deviations of polygenic
(
EBV for standard and optimal terminal QTL selection. Starting frequencies are 0.1
for both QTL.
Linkage had a limited effect on QTL frequencies for standard selection
(Fig. 3), although rates of fixation were slightly reduced with linkage because
favorable alleles in repulsion phase were not released as quickly. For optimal
selection, both QTL approached fixation regardless of the degree of linkage.
However, increases in QTL frequencies were smaller with linkage than without
linkage in early generations. The reason for the different effect of linkage for
standard versus optimal QTL selection is explored in the following section.
3.3. QTL haplotype frequencies
Changes in frequencies for three QTL haplotypes are shown in Figure 4a for
standard QTL selection and in Figure 4b for optimal terminal QTL selection.
Standard QTL selection resulted in rapid fixation to a frequency of one of
the favorable haplotype (A1 B1 ) and to zero of the unfavorable haplotype
(A2 B2 ). Standard QTL selection initially increased the frequency of the
haplotype in the repulsion phase (A2 B1 ) before reducing its frequencies to zero.
Linkage had a limited impact on these trends, although the move to fixation
of the favorable haplotype was slightly slower and the rise and subsequent
decline of frequencies for repulsion phase haplotypes were slightly greater
with linkage. This is caused by the slower break-up of linked QTL in repulsion
phase.
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1.0
(a) Standard
0.9
0.8
A1B1
Frequency
0.7
A2B1
0.6
A2B2
0.5
0.4
0.3
0.2
0.1
0.0
0
1
2
3
4
5
6
7
8
9
10
8
9
10
Generation
1.0
Frequency
0.9
A1B1
(b) Optimal
0.8
A2B1
0.7
A2B2
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0
1
2
3
4
5
6
7
Generation
Figure 4. Frequencies of selected QTL haplotypes for two QTL with recombination
rates equal to 0.5 (- - - -), 0.2 (
), or 0.05 (
), and effects equal to 1.0 (A) and
1.5 (B) standard deviations of polygenic EBV, for (a) standard and (b) optimal terminal
QTL selection. Starting frequencies of the favorable alleles for each QTL (A1 and B1 )
are 0.1.
Quite different trends in haplotype frequencies were observed for optimal
QTL selection (Fig. 4b). Frequencies of the favorable and unfavorable haplotypes also increased and decreased, respectively, but much more gradually than for standard QTL selection. Linkage had a limited impact on the
increasing trend for the favorable haplotype but resulted in a slower decline
in the frequency of the unfavorable haplotype. In contrast to standard QTL
selection, frequencies for the repulsion phase haplotype were greatly impacted
by linkage under optimal selection; without linkage, the frequency of the
Optimal selection on two linked QTL
183
repulsion phase haplotype increased for the first five generations, after which
it decreased towards zero. With tight linkage, however, the frequency of the
repulsion phase haplotype was nearly constant for the first eight generations,
followed by a gradual decline towards zero. With tight linkage, repulsion phase
haplotypes do not break up through recombination and must be selected against
in order to eventually fix the favorable QTL alleles. Thus, the initial increase
in frequency of the repulsion phase haplotype with standard QTL selection
(Fig. 4b) represents a wasted selection effort because it has to be selected
against in later generations. This wastage is avoided by optimal QTL selection
by minimizing the increase in frequency of such haplotypes. This ability
to minimize wasted selection effort also explains why the benefit of optimal
QTL over standard QTL and phenotypic selection is greater with tight linkage
(Tab. I). Most of these effects disappeared, however, with moderate linkage,
which behaved similar to unlinked QTL under optimal selection (Fig. 4b).
3.4. Fractions selected
The effect of linkage on selection strategies is further illustrated in Figure 5,
which shows fractions selected from specific QTL genotypes with tight (r =
0.05) and no linkage for standard (Fig. 5a) and optimal terminal QTL selection
(Fig. 5b). The number of favorable and unfavorable haplotypes and alleles
that is present in each genotype is also indicated. For brevity, only the results
for generations 0, 4, and 8 are shown. The genotypes that are not depicted
exhibited similar trends as the presented genotypes, based on the number of
favorable and unfavorable haplotypes they contain.
For standard QTL selection, fractions selected from each genotype decreased
with a declining QTL value of the genotype (Fig. 5a), as expected. The selected
fractions decreased over generations because of the increasing frequency of the
favorable alleles. Linkage had no impact on the fractions selected in the initial
generation, as expected, and limited impact in later generations, which was
primarily caused by differences in genotype frequencies.
For optimal QTL selection (Fig. 5b), differences in fractions selected
between genotypes were not as large as for standard selection and linkage had
a much greater impact. Without linkage, the selected fractions decreased with
and were almost entirely determined by the number of favorable alleles, regardless of the linkage phase and the difference in the effect of the alleles between
the two QTL. This is consistent with the observed similarity in frequency
trends between the two QTL (Fig. 1). With tight linkage, however, the selected
fractions were determined primarily by the number of favorable haplotypes,
with a slight negative emphasis on the number of unfavorable haplotypes.
For example, fractions selected were very similar for genotypes A1 A1 B1 B2
(1 favorable and 0 unfavorable haplotypes) and A1 A2 B1 B2 (1 favorable and
1 unfavorable haplotype). Genotypes with no favorable haplotypes were
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1
Standard selection
0.9
Unlinked
Linked
Fraction selected
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
4
8
0
4
8
0
4
8
0
4
8
0
4
8
0
4
8
Generation
Genotype
A1A1B1B1
No. favorable
haplotypes
A1A1B1B2
A1A2B1B2
A1A2B2B1
A1A2B2B2
A2A2B2B2
2
1
1
0
0
0
No unfavorable
haplotypes
0
0
1
0
1
2
No. favorable
alleles
3
2
2
1
0
4
Optimal selection
1
Unlinked
Linked
0.9
Fraction selected
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
4
8
0
4
8
0
4
8
0
4
8
0
4
8
0
4
8
Generation
Figure 5. Proportions retained for breeding and haplotype characteristics for selected
QTL genotypes in selected generations for standard and optimal terminal QTL selection for two unlinked () or linked () QTL, with effects equal to 1.0 (A) and 1.5 (B)
standard deviations of polygenic EBV. Starting frequencies of the favorable alleles for
each QTL (A1 and B1 ) are 0.1.
selected with the lowest emphasis; and, for those genotypes the fractions selected decreased slightly with increasing number of unfavorable haplotypes. With
tight linkage, greater fractions of the most unfavorable genotype A2 A2 B2 B2
were selected than without linkage. Consequently, the decline in frequency
of the A2 B2 haplotype observed with linkage in Figure 4b was less rapid than
without linkage. Moderate linkage (r = 0.2, results not shown) had results that
were intermediate but closer to the situation with no linkage.
185
Optimal selection on two linked QTL
Table III. Proportions selected and average genotypic values (in brackets) assigned to
individual QTL genotypes under optimal QTL selection to maximize terminal response
after ten generations for two unlinked (r = 0.5) or linked (r = 0.05) QTL with effects
equal to 1.0 (A) and 1.5 (B) standard deviations of polygenic EBV. Starting frequencies
of the favorable alleles for each QTL (A1 and B1 ) are 0.1.
Generation
1
3
5
7
9
1
3
5
7
9
A1 A1
(+1.0)
QTL genotype
(Effect of QTL genotype on phenotype)
B1 B1
A1 A2
B1 B2
A2 A2
(+1.5)
(0)
(0)
(−1.0)
B2 B2
(−1.5)
0.45
(0.28)
0.28
(0.09)
0.27
(0.12)
0.25
(0.18)
0.24
(n/a) ∗∗
0.46
(0.28)
0.28
(0.09)
0.27
(0.13)
0.25
(0.20)
0.24
(0.45)
Unlinked QTL (r = 0.5)
0.34
0.35
(0) ∗
(0)
0.25
0.25
(0)
(0)
0.23
0.23
(0)
(0)
0.2
0.2
(0)
(0)
0.16
0.14
(0)
(0)
0.16
(−0.60)
0.16
(−0.36)
0.13
(−0.41)
0.09
(−0.63)
0.01
(n/a)
0.16
(−0.62)
0.15
(−0.37)
0.13
(−0.45)
0.07
(−0.75)
0.0
(n/a)
0.32
(0.17)
0.29
(0.12)
0.28
(0.12)
0.28
(0.17)
0.27
(n/a)
0.33
(0.19)
0.29
(0.12)
0.28
(0.12)
0.27
(0.17)
0.26
(n/a)
Linked QTL (r = 0.05)
0.26
0.27
(0)
(0)
0.25
0.25
(0)
(0)
0.25
0.24
(0)
(0)
0.23
0.23
(0)
(0)
0.16
0.15
(0)
(0)
0.19
(−0.23)
0.17
(−0.25)
0.15
(-0.33)
0.1
(−0.53)
0.0
(n/a)
0.18
(−0.25)
0.17
(−0.25)
0.15
(-0.34)
0.09
(−0.56)
0.0
(n/a)
∗
Used as reference genotype.
Could not be computed because selected proportion was zero for one of the
genotypes.
∗∗
Fractions selected from genotypes for each of the two QTL, averaged over
the other QTL, are given in Table III for optimal terminal QTL selection for
two unlinked or tightly linked QTL. The selected fractions were very similar
for the two QTL, regardless of linkage, which is consistent with the similar
trends in QTL frequencies observed in Figure 3.
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J.C.M. Dekkers et al.
3.5. Optimal QTL genotype values
Table IV shows genotype values assigned to selected QTL genotypes under
optimal terminal selection on unlinked QTL, which were derived following
equation (2). Optimal genotype values generally declined with the number of
favorable alleles. Double heterozygotes (A1 A2 B1 B2 ) were, however, assigned
a slightly greater value than the genotypes that were homozygous for alternate
alleles (e.g. A1 A1 B2 B2 ), which is most likely associated with gametic phase
disequilibrium among the QTL and between the QTL and polygenes.
Optimal single locus genotype values are given in Table III. Similar to
proportions selected, optimal single locus genotype values were nearly equal
for the two QTL, indicating that equal emphasis is placed on both loci. For
both QTL the value of the heterozygote was closer to the value of the favorable
homozygote. This was also observed by Dekkers and van Arendonk [5] for one
QTL and relates to reducing gametic phase disequilibrium between the QTL
and polygenes (see later).
If selection treats the two QTL loci as independent, the optimal genotype
value of a bilocus QTL genotype will equal the sum of the optimal single locus
genotype values. For example, the genotype value of genotype A1 A2 B1 B2 will
equal the single locus value for A1 A2 plus the single locus value for B1 B2 .
Comparing the optimal single locus genotype values of Table III to the bilocus
genotype values of Table IV shows that this independence does not hold for
optimal selection, even if the QTL are unlinked.
3.6. Gametic phase disequilibrium
The extent of gametic phase disequilibrium generated by optimal terminal
and standard QTL selection between the two QTL is presented in Figure 6a for
unlinked and tightly linked QTL. Standard QTL selection resulted in increasing negative disequilibrium between the two QTL over generations (Fig. 6a).
Negative disequilibrium here indicates that individuals with a more favorable
genotype for one QTL tended to have a more unfavorable genotype for the other
QTL, which is caused by an excess frequency of repulsion phase haplotypes.
Linkage increased the amount of disequilibrium for standard QTL selection
because repulsion phase haplotypes recombined less frequently to break up the
disequilibrium (Fig. 4a).
Disequilibrium between the QTL was small but negative for optimal selection without linkage (Fig. 6a). With tight linkage, however, optimal selection
resulted in a large positive disequilibrium between the QTL. Positive disequilibrium indicates that individuals that had a favorable genotype for one QTL
also tended to have a favorable genotype at the other QTL. The large positive
disequilibrium for optimal selection resulted from its ability to maintain the
frequency of repulsion phase haplotypes at low frequencies (Fig. 4b).
Table IV. Optimal genotype values assigned to selected QTL genotypes under optimal selection on two unlinked QTL (A and B) with
effects equal to 1.0 (A) and 1.5 (B) standard deviations of polygenic EBV, to maximize terminal response at the end of ten generations.
Starting frequencies of the favorable alleles for each QTL (A1 and B1 ) are 0.1.
QTL genotype
(number of favorable alleles/effect of QTL genotype on phenotype)
A1 A1 ∗
A1 A1
A1 A2
B1 B1
B1 B2
B1 B1
(4/ + 2.5) (3/ + 1.0) (3/ + 1.5)
A1 A2
B1 B2
(2/0)
1
0.42
0.22
0.23
0 ∗∗
2
0.14
0.06
0.06
3
0.12
0.06
4
0.15
5
A1 A2
B2 B1
(2/0)
A1 A1
A2 A2
A2 A2
A1 A2
A2 A2
B2 B2
B1 B1
B1 B2
B2 B2
B2 B2
(2/ − 0.5) (2/ + 0.5) (1/ − 1.0) (1/ − 1.5) (0/ − 2.5)
0.00
−0.16
−0.14
−0.43
−0.45
−1.10
0
−0.06
−0.21
−0.20
−0.32
−0.33
−0.79
0.05
0
−0.03
−0.18
−0.17
−0.30
−0.31
−0.77
0.08
0.08
0
−0.02
−0.18
−0.17
−0.32
−0.34
−0.86
0.19
0.11
0.11
0
−0.01
−0.19
−0.17
−0.37
−0.40
−1.04
6
0.25
0.15
0.15
0
0.00
−0.23
−0.19
−0.46
−0.51
−1.38
7
0.34
0.20
0.21
0
0.00
−0.32
−0.24
−0.64
−0.75
−2.16
8
0.48
0.29
0.31
0
0.00
−0.66
−0.37
−1.08
−1.49
n/a ∗∗∗
9
0.87
0.52
0.58
0
0.00
n/a
−0.79
−2.32
n/a
n/a
10
2.07
0.75
1.23
0
−0.10
0.39
−0.87
−1.16
n/a
−0.59
Optimal selection on two linked QTL
Generation
∗
The first allele refers to that inherited from the sire.
Genotype A1 A2 B1 B2 was used as the reference genotype from which all other values were deviated.
∗∗∗
Values could not be computed because fraction selected was zero for that genotype.
∗∗
187
188
J.C.M. Dekkers et al.
(a) Between QTL
0.80
0.60
Optimal
Disequilibrium
0.40
0.20
0.00
-0.20
Optimal
-0.40
-0.60
Standard
-0.80
-1.00
0
1
2
3
4
5
6
Generation
7
8
9
10
0.00
(b) Between QTL and polygenes
Disequilibrium
-0.02
-0.04
Optimal
-0.06
-0.08
Standard
-0.10
-0.12
0
1
2
3
4
5
6
Generation
7
8
9
10
Figure 6. Disequilibrium between two unlinked (- - - -) or linked (r = 0.05) QTL
(
) QTL for standard (•) and optimal (◦) QTL selection.
Both standard and optimal QTL selection resulted in negative disequilibrium between the QTL and polygenes for all cases (Fig. 6b), indicating
that individuals with more favorable QTL genotypes tended to have lower
Optimal selection on two linked QTL
189
polygenic breeding values. During the first five generations, correlations of
QTL with polygenic values were twice as large for standard than for optimal
QTL selection. In later generations the correlation reduced towards zero for
standard QTL selection, as the QTL were moved to fixation. Tight linkage
between the QTL slightly increased the extent of disequilibrium for standard QTL selection. For optimal selection, tight linkage resulted in slightly
greater disequilibrium in early and late generations but in less disequilibrium
in intermediate generations.
4. DISCUSSION
This study was focused on the evaluation of optimal strategies for simultaneous selection on two QTL, in comparison to standard QTL and phenotypic selection, and on the effect of linkage between the QTL on these
strategies. Optimal strategies were derived using the methodology and algorithm developed by Chakraborty et al. [2].
Although only a limited number of scenarios was investigated, the results
indicate that the benefit of optimal QTL selection over phenotypic selection is
greater when the identified effects are distributed over multiple QTL. This holds
for both terminal response (Tab. I) and for cumulative discounted response
(Tab. II). Phenotypic selection has more difficulty in selecting for genes of
small effect and, therefore, the benefit of being able to directly select on such
genes is greater than for genes of large effect. The benefit of optimal over
standard QTL selection was, however, greater for one QTL than for two QTL
(Tabs. I and II), unless the QTL were tightly linked and the objective was to
maximize cumulative discounted response (Tab. II). Tight linkage between the
two QTL increased the advantage of optimal over standard QTL selection but
moderate linkage (r = 0.2) had a limited impact and behaved very similar to
selection on unlinked QTL.
Upon comparing optimal QTL selection with and without linkage between
the two QTL, the results presented clearly demonstrate that the optimal strategy
is able to take into account the ability for haplotypes that are in repulsion phase
to recombine in future generations. This resulted in even greater response to
selection with tight linkage than without linkage (Tab. I). This is in contrast
to phenotypic and standard QTL selection, which wasted selection effort by
selecting for haplotypes that are in repulsion phase in early generations. For
these strategies, linkage resulted in slightly lower genetic gain, although the
differences were not large. Most of these effects, however, disappeared with
moderate linkage (r = 0.2), which behaved similar to selection on unlinked
QTL because there is sufficient opportunity for QTL to recombine.
Only additive QTL were considered here, which is the most favorable
situation for standard QTL selection. Greater differences between optimal
190
J.C.M. Dekkers et al.
and standard QTL selection are expected for QTL with dominance, as was
demonstrated by Dekkers and Chakraborty [6] for the one QTL case. Epistasis
between QTL may further increase the benefit of optimizing selection. Opportunities to capitalize on complex genetic inheritance, i.e. multiple QTL, linkage,
epistasis, etc., will, however, be limited by the ability to accurately estimate
associated parameters. In the present study, all parameters were assumed
known without error. Estimation errors, which will be larger for complex
genetic models, will limit the benefit of optimal selection. Nevertheless,
these findings illustrate the potential of the optimization methods developed by
Chakraborty et al. [3] to develop optimal selection strategies to capitalize on
the complex genetic systems and interactions that are being uncovered through
molecular genetic technology.
Several interesting features of optimal terminal selection were observed in
Figures 2, 3, and 5 that deserve further discussion. First, both QTL reached
near fixation by the end of the planning horizon (Fig. 2). Second, selection
emphasis differed little between the two QTL when starting frequencies were
equal, despite a difference in effects between QTL, as is evident from Figure 3
by the overlapping trends in QTL frequencies, and from Table III from the
nearly equal fractions selected and equal optimal genotype effects for the two
QTL. Third, trends in QTL frequencies were nearly linear when the QTL were
unlinked, even when the two QTL started at different frequencies (Fig. 2).
To understand the impact of these findings it is beneficial to reformulate
the selection objective of maximizing genetic level in the final generation to
maximizing the polygenic value in the final generation subject to fixation of both
QTL in the final generation. This equivalence of objectives holds for the QTL
effects considered here and is based on the finding that in the optimal strategy
both QTL were essentially fixed in the final generation. Thus, the assertion
here is that the reformulated objective would lead to a similar selection strategy
as the objective of maximizing terminal gain that was implemented here. From
the reformulated objective, it is indeed evident that there is no advantage to
putting more emphasis on the larger QTL if both start at the same frequency.
After all, the selection effort that is required to move the favorable allele to
fixation is equal for both QTL and does not depend on the effect of the QTL.
The near linear trends that were observed with different starting frequencies
(Fig. 2) further assert this principle. Thus, when the objective is to maximize
terminal response, emphasis on a QTL is primarily determined by its starting
frequency and not by the magnitude of its effect. It is important to note that
the size of the QTL effects will be important when the objective is to maximize
cumulative discounted response, as illustrated by the more rapid fixation of the
QTL with larger effects (Fig. 2).
The latter also confirms the validity of the strategy of putting equal emphasis
on each QTL when the objective is to reach fixation for all QTL as quickly as
Optimal selection on two linked QTL
191
possible and all QTL start at the same frequency. This strategy was adopted in
a simulation study by Hospital et al. [10], in which selection was exclusively
on the QTL and all QTL had the same starting frequency. It is clear that
also in that case the selection objective is not enhanced by giving QTL with
greater genetic values more emphasis. Based on this argument, Hospital
et al. [10] selected on the expected number of favorable QTL alleles contained
by an individual. The results presented here indicate that this strategy would
indeed be optimal for unlinked QTL because the selected fractions (Fig. 5)
and optimal values for each genotype (Tab. IV) decreased with the number
of favorable alleles. For tightly linked QTL, however, the optimal emphasis
depended primarily on the number of favorable haplotypes, rather than on
the number of favorable alleles, with some consideration of the number of
unfavorable haplotypes (Fig. 5). Further work is needed to extend this result to
multiple groups of linked QTL, which was the situation considered by Hospital
et al. [10].
Hospital et al. [10] selected exclusively on markers with no consideration
of other polygenic effects. Including residual polygenic effects in the selection
decision requires consideration of not only the relative emphasis between QTL
but also the emphasis on QTL versus polygenes. This was the situation
considered here. The results presented in Tables III and IV for terminal
selection on unlinked QTL show that, although the effects of the QTL on
phenotype were additive and independent, the values assigned to the QTL
under optimal selection behaved in a non-additive manner. Specifically, the
average value assigned by the optimal strategy to the heterozygotes for each
QTL was not central to that of the two homozygotes (Tab. III). In addition,
when comparing optimal bilocus genotype values of Table IV to the optimal
single locus genotype values of Table III, the optimal genotype values at the
two QTL were not additive. Both types on non-additivity, as observed under
optimal selection, are likely related to gametic phase disequilibrium among the
QTL and between the QTL and polygenes. It is well known that selection on an
aggregate objective induces negative associations between components of the
aggregate (i.e. the two QTL and polygenes), which reduces response to selection
in future generations [2]. Even for, unlinked loci, negative associations and
their effects on response to selection will persist over multiple generations [2].
Optimal selection is able to anticipate the impact of negative associations on
future responses to selection and, therefore, reduces the build-up of negative
disequilibrium, as illustrated in Figures 6a and 6b. This is accomplished in part
by a reducing emphasis on the QTL and in part by putting differential emphasis
on alternate QTL genotypes. The latter results in the observed non-additivity
of optimal QTL genotype values. It is not clear how important the assignment
of non-additive effects under optimal selection is; responses may be robust to
moderate changes in optimal QTL values.
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J.C.M. Dekkers et al.
ACKNOWLEDGEMENTS
Financial support from the Pig Improvement Company and the NRI Competitive Grants Program/USDA (award no. 98-35205-6736) is greatly appreciated.
This is Journal Paper No. J-19214 of the Iowa Agriculture and Home Economics Experiment Station, Ames, Iowa, USA (Project No. 3456) and supported
by the Hatch Act and State of Iowa Funds. During her stay at ISU, L. Moreau
was funded by INRA.
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